For improvement, two elements play a role, skill and condensed knowledge.
Skill is something that can become better by just exercise it automatic and unconscious, “like a madman”/Troyis-style. Skill is about manipulating the pieces and the squares in the mind.
Thinking is a slow conscious process which can yield knowledge. To make knowledge accessible in a practical way, it has to be “ready knowledge”. For chess, the most practical way to express knowledge is a diagram, complete with arrows and colored squares, if needed. I will use the term geometrical pattern, although that might not be quite correct technically. By learning these geometrical patterns by heart, they will become available for direct use.
When your progress is stalling at a certain exercise, you can go in two ways. Either you try to acquire more condensed knowledge, or you look for more basic skills. The stalling itself means that your skill for that exercise has reached it's current maximum.
The road of acquiring more condensed knowledge is subject to the law of diminishing returns. It becomes more difficult over time to find geometrical patterns that have a high enough frequency of occurrence to be of any practical use. When you have done some serious work on tactical exercises, you are probably well versed in the most common patterns. In a strategical underdeveloped game like Troyis, you can gain high results by inventing new patterns. But the path of chess tactics is already well trodden.
So it is skill time!
What exercises do we have?
FAC is essentially about the chain Attacker – Attacking square – Target.
Further we have FAAP – Find All Attacked pieces. This is even more elemental than FAC, since the attacking square is left out. I will give it a try.
There are no specific exercises for restraint. M1-h is the first exercise with a high “restrain-dose”, but since it is a compound exercise, you are never able to train your restrain-skill at high speed, since the other underdeveloped skills, which you need for the other parts of the M1-h-exercise, will slow you down.
In order to train restrain, a subset of all M1-h can be used, with solely white pieces and a black king. This rules out the subtasks “blocking the attack” and “protecting the attacking square”, since black pieces are needed for that.
When doing tactical exercises I found that lack of restrain-vision was my main bottleneck when trying to visualize-II future positions.
For blocking the line of attack by interposing a piece doesn't exist a specific training, as far as I know. Blocking is a subtask of M1-h, so M1-h can be used to train it, with the same limitations as restraint.
There is a specific exercise FADP – Find All Defended Pieces, that might represent a basic skill. For the sake of elimination, it is good to master all these basic exercises to the max. M1-h has Find All Defended Squares as a subtask. In the form of “Is this attacking square I intend to move to protected?”
Tomasz' initial speed at exercises like M1-e, M1-h and FAC is far greater than mine. That suggests he masters a subskill that is used in all these exercises. Maybe is willing to share what that is and how he trained it? ;)
For the time being I will have a look if the subtask Find All Attacked Pieces is of any influence on Find All Checks.